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Gap Amplification in PCPs Using Lazy Random Walks

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 4051)

Abstract

We show an alternative implementation of the gap amplification step in Dinur’s [4] recent proof of the PCP theorem. We construct a product G t of a constraint graph G, so that if every assignment in G leaves an ε-fraction of the edges unsatisfied, then in G t every assignment leaves an Ω()-fraction of the edges unsatisfied, that is, it amplifies the gap by a factor Ω(t). The corresponding result in [4] showed that one could amplify the gap by a factor \(\Omega(\sqrt{t})\). More than this small quantitative improvement, the main contribution of this work is in the analysis. Our construction uses random walks on expander graphs with exponentially distributed length. By this we ensure that some random variables arising in the proof are automatically independent, and avoid some technical difficulties.

Keywords

  • Random Walk
  • Original Graph
  • Constraint Graph
  • Product Construction
  • Expander Graph

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References

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  4. Dinur, I.: The PCP Theorem by Gap Amplification. ECCC TR05-046, http://eccc.uni-trier.de/eccc-reports/2005/TR05-046

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  6. O’Donnell, R., Guruswami, V.: Course notes of CSE 533: The PCP Theorem and Hardness of Approximation, http://www.cs.washington.edu/education/courses/533/05au/

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© 2006 Springer-Verlag Berlin Heidelberg

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Radhakrishnan, J. (2006). Gap Amplification in PCPs Using Lazy Random Walks. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_10

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  • DOI: https://doi.org/10.1007/11786986_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35904-3

  • Online ISBN: 978-3-540-35905-0

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